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This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
You're on a roll — Step 1: Evaluate [3]63^3. When taking the n-th root of a number raised to the n-th power, the root and the power cancel each other out. [3]63^3 = 63 63 Step 2: Simplify sqrt((297)/(64)). First, use the property sqrt((a)/(b)) = sqrt(a)sqrt(b): sqrt((297)/(64)) = sqrt(297)sqrt(64) Next, simplify the numerator sqrt(297). Find the largest perfect square factor of 297. 297 = 9 × 33. So, sqrt(297) = sqrt(9 × 33) = sqrt(9) × sqrt(33) = 3sqrt(33). Now, simplify the denominator sqrt(64): sqrt(64) = 8 Combine the simplified numerator and denominator: 3sqrt(33)8 3sqrt(33)8 Step 3: Express in power form: (5)/(7) × (5)/(7) × (5)/(7) × (5)/(7). When a base is multiplied by itself multiple times, it can be written in power form where the base is the number being multiplied and the exponent is the number of times it is multiplied. The base is (5)/(7) and it is multiplied 4 times. ((5)/(7))^4 ((5)/(7))^4 Step 4: Evaluate ((3)/(11))^-1 × ((3)/(11))^2 × ((3)/(11))^1 × ((3)/(11))^-4. When multiplying terms with the same base, add their exponents. The base is (3)/(11). Add the exponents: -1 + 2 + 1 + (-4). -1 + 2 + 1 - 4 = 1 + 1 - 4 = 2 - 4 = -2 So the expression simplifies to: ((3)/(11))^-2 To evaluate a term with a negative exponent, use the rule a^-n = (1)/(a^n) or ((a)/(b))^-n = ((b)/(a))^n: ((3)/(11))^-2 = ((11)/(3))^2 Now, apply the exponent to both the numerator and the denominator: (11^2)/(3^2) = (11 × 11)/(3 × 3) = (121)/(9) (121)/(9) What's next?